This type of separating device is known, for example, from DE 41 3 792 A1. In separating devices of this type, a tubular web is advanced at a certain speed by a pulling mechanism. While the tubular web is being pulled it runs through a tearing mechanism positioned in the direction of travel, without the rollers or cylinders of the tearing mechanism, or the belts guided by them, engaging with the tubular web. If a tubular web is being separated into sections along a perforation running perpendicular to the length of the tubular web, the rollers or cylinders of the tearing mechanism are positioned against the tubular web and are driven at a greater circumferential speed than the rollers or cylinders of the pulling mechanism, to thereby tear off the tubular section. Due to this speed differential, a tractive force is exerted on the entire following portion of the web during tearing. As long as the bridge portions of the perforation have not yet been torn, the tractive force continues into the pulling mechanism. In the known separating devices the facing rollers or cylinders of the pulling mechanism are positioned in staggered fashion. The tubular web thus runs in serpentine fashion between these rollers or cylinders. Practical operation has shown that the pressure exerted on the tubular web by the belts running over the rollers or cylinders is not sufficient to hold the tubular web in position when the tearing mechanism is engaged and, as described above, when the tractive force is exerted on the following portion of the tubular web. This results in undesired slippage, which unfavorably affects the tearing process and may cause strips to form in the tubular web due to the web sliding over the belts.